How do you find the exact values of cos(5pi/12) using the half angle formula?

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Alan P. Share
Mar 31, 2017

Answer:

#cos((5pi)/12) = (sqrt(2-sqrt(3)))/2#

Explanation:

By the half angle formula:
#color(white)("XXXX")##cos(theta/2) = +-sqrt((1+cos(theta))/2)#

If #theta/2 = (5pi)/12#
#color(white)("XXXX")#then #theta = (5pi)/6#

Note that #(5pi)/6# is a standard angle in quadrant 2 with a reference angle of #pi/6#
so #cos((5pi)/6) = -cos(pi/6) = -sqrt(3)/2#

Therefore
#color(white)("XXXX")cos((5pi)/12) = +- sqrt((1-sqrt(3)/2)/2)#

#color(white)("XXXXXXXXXXX")=+-sqrt(((2-sqrt(3))/2)/2)#

#color(white)("XXXXXXXXXXX")=+-sqrt((2-sqrt(3))/4)#

#color(white)("XXXXXXXXXXX")=+-sqrt(2-sqrt(3))/2#

Since #(5pi)/12 < pi/2#
#color(white)("XXXX")##(5pi)/12# is in quadrant 1
#color(white)("XXXX")##rarr cos((5pi)/12)# is positive
#color(white)("XXXX")##color(white)("XXXX")##color(white)("XXXX")#(the negative solution is extraneous)

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